I think it was summed up best by a very Scottish young Scotsman I was chatting to on the beach one Sunday afternoon, shortly after I had returned home after many years of living away. I remarked that his home town, Edinburgh, was just about my favourite city in Europe, and he said "Och aye, laddie -- but Sydney is Edinburgh with salt and pepper on it!"
As Matilda says, it is a very beautiful city, blessed with an incomparable setting, a nice almost-subtropical climate and relaxed and friendly people, and you'll find your interests very well catered for here. But I would recommend that if you're only here for a couple of months you base yourself fairly close to the central city if you can. Sydney is a vast metropolis, but it is also a very centralised one. I'd draw a line around the CBD, say from Paddington in the east to Newtown in the south and Balmain in the west, the old Victorian-era suburbs.
That will give you the best of Sydney on your doorstep -- the theatres, the concert halls, the best food (and you'll eat here as well as anywhere in the world) the Harbour and the beaches -- and you will then find the public transport system will serve you well. (The gripes are really mainly about poor service to the far-flung suburbs.) Jump on a train at Central and you'll find some of the world's loveliest wilderness less than two hours away; but if you have to start the trip with a long slow ride into town it rubs the shine off.
A bit of background reading for you:
http://en.wikipedia.org/wiki/SydneyA bit impersonal, but (with its links) it gives you a good and fairly accurate overview.
http://www.sydneyaustralia.com/en/The New South Wales government's information site and a bit of a hard sell, but well packed with links to useful info
http://sydneynearlydailyphot.blogspot.com/Sally's excellent daily photo blog, well worth spending a couple of hours on. She gives a nice feeling of what life here is like.